Chapter 6 Linear Transformations 6.1 Introduction to Linear Transformations 6.2 The Kernel and Range of a Linear Transformation 6.3 Matrices for Linear Transformations 6.4 Transition Matrices and Similarity Elementary Linear Algebra R. Larsen et al. (6 Edition)

6.1 Introduction to Linear Transformations Function T that maps a vector space V into a vector space W: V: the domain of T W: the codomain of T Elementary Linear Algebra: Section 6.1, pp.361-362

If v is in V and w is in W such that Image of v under T: If v is in V and w is in W such that Then w is called the image of v under T . the range of T: The set of all images of vectors in V. the preimage of w: The set of all v in V such that T(v)=w. Elementary Linear Algebra: Section 6.1, p.361

Ex 1: (A function from R2 into R2 ) (a) Find the image of v=(-1,2). (b) Find the preimage of w=(-1,11) Sol: Thus {(3, 4)} is the preimage of w=(-1, 11). Elementary Linear Algebra: Section 6.1, p.362

Linear Transformation (L.T.): Elementary Linear Algebra: Section 6.1, p.362

(1) A linear transformation is said to be operation preserving. Notes: (1) A linear transformation is said to be operation preserving. Addition in V Addition in W Scalar multiplication in V Scalar multiplication in W (2) A linear transformation from a vector space into itself is called a linear operator. Elementary Linear Algebra: Section 6.1, p.363

Ex 2: (Verifying a linear transformation T from R2 into R2) Pf: Elementary Linear Algebra: Section 6.1, p.363

Therefore, T is a linear transformation. Elementary Linear Algebra: Section 6.1, p.363

Ex 3: (Functions that are not linear transformations) Elementary Linear Algebra: Section 6.1, p.363

Notes: Two uses of the term “linear”. (1) is called a linear function because its graph is a line. (2) is not a linear transformation from a vector space R into R because it preserves neither vector addition nor scalar multiplication. Elementary Linear Algebra: Section 6.1, p.364

Identity transformation: Zero transformation: Identity transformation: Thm 6.1: (Properties of linear transformations) Elementary Linear Algebra: Section 6.1, p.365

Ex 4: (Linear transformations and bases) Let be a linear transformation such that Find T(2, 3, -2). Sol: (T is a L.T.) Elementary Linear Algebra: Section 6.1, p.365

Ex 5: (A linear transformation defined by a matrix) The function is defined as Sol: (vector addition) (scalar multiplication) Elementary Linear Algebra: Section 6.1, p.366

Thm 6.2: (The linear transformation given by a matrix) Let A be an mn matrix. The function T defined by is a linear transformation from Rn into Rm. Note: Elementary Linear Algebra: Section 6.1, p.367

Ex 7: (Rotation in the plane) Show that the L.T. given by the matrix has the property that it rotates every vector in R2 counterclockwise about the origin through the angle . Sol: (polar coordinates) r： the length of v ：the angle from the positive x-axis counterclockwise to the vector v Elementary Linear Algebra: Section 6.1, p.368

 +：the angle from the positive x-axis counterclockwise to r：the length of T(v)  +：the angle from the positive x-axis counterclockwise to the vector T(v) Thus, T(v) is the vector that results from rotating the vector v counterclockwise through the angle . Elementary Linear Algebra: Section 6.1, p.368

The linear transformation is given by Ex 8: (A projection in R3) The linear transformation is given by is called a projection in R3. Elementary Linear Algebra: Section 6.1, p.369

Ex 9: (A linear transformation from Mmn into Mn m ) Show that T is a linear transformation. Sol: Therefore, T is a linear transformation from Mmn into Mn m. Elementary Linear Algebra: Section 6.1, p.369

Keywords in Section 6.1: function: 函數 domain: 論域 codomain: 對應論域 image of v under T: 在T映射下v的像 range of T: T的值域 preimage of w: w的反像 linear transformation: 線性轉換 linear operator: 線性運算子 zero transformation: 零轉換 identity transformation: 相等轉換

6.2 The Kernel and Range of a Linear Transformation Kernel of a linear transformation T: Let be a linear transformation Then the set of all vectors v in V that satisfy is called the kernel of T and is denoted by ker(T). Ex 1: (Finding the kernel of a linear transformation) Sol: Elementary Linear Algebra: Section 6.2, p.375

Ex 2: (The kernel of the zero and identity transformations) (a) T(v)=0 (the zero transformation ) (b) T(v)=v (the identity transformation ) Ex 3: (Finding the kernel of a linear transformation) Sol: Elementary Linear Algebra: Section 6.2, p.375

Ex 5: (Finding the kernel of a linear transformation) Sol: Elementary Linear Algebra: Section 6.2, p.376

Elementary Linear Algebra: Section 6.2, p.377

Thm 6.3: (The kernel is a subspace of V) The kernel of a linear transformation is a subspace of the domain V. Pf: Note: The kernel of T is sometimes called the nullspace of T. Elementary Linear Algebra: Section 6.2, p.377

Ex 6: (Finding a basis for the kernel) Find a basis for ker(T) as a subspace of R5. Elementary Linear Algebra: Section 6.2, p.377

Sol: Elementary Linear Algebra: Section 6.2, p.378

Range of a linear transformation T: Corollary to Thm 6.3: Range of a linear transformation T: Elementary Linear Algebra: Section 6.2, p.378

Thm 6.4: (The range of T is a subspace of W) Pf: Elementary Linear Algebra: Section 6.2, p.379

Notes: Corollary to Thm 6.4: Elementary Linear Algebra: Section 6.2, p.379

Ex 7: (Finding a basis for the range of a linear transformation) Find a basis for the range of T. Elementary Linear Algebra: Section 6.2, p.379

Sol: Elementary Linear Algebra: Section 6.2, pp.379-380

Rank of a linear transformation T:V→W: Nullity of a linear transformation T:V→W: Note: Elementary Linear Algebra: Section 6.2, p.380

Thm 6.5: (Sum of rank and nullity) Pf: Elementary Linear Algebra: Section 6.2, p.380

Ex 8: (Finding the rank and nullity of a linear transformation) Sol: Elementary Linear Algebra: Section 6.2, p.381

Ex 9: (Finding the rank and nullity of a linear transformation) Sol: Elementary Linear Algebra: Section 6.2, p.381

One-to-one: one-to-one not one-to-one Elementary Linear Algebra: Section 6.2, p.382

(T is onto W when W is equal to the range of T.) Elementary Linear Algebra: Section 6.2, p.382

Thm 6.6: (One-to-one linear transformation) Pf: Elementary Linear Algebra: Section 6.2, p.382

Ex 10: (One-to-one and not one-to-one linear transformation) Elementary Linear Algebra: Section 6.2, p.382

Thm 6.7: (Onto linear transformation) Thm 6.8: (One-to-one and onto linear transformation) Pf: Elementary Linear Algebra: Section 6.2, p.383

Ex 11: Sol: T:Rn→Rm dim(domain of T) rank(T) nullity(T) 1-1 onto (a)T:R3→R3 3 Yes (b)T:R2→R3 2 No (c)T:R3→R2 1 (d)T:R3→R3 Elementary Linear Algebra: Section 6.2, p.383

Thm 6.9: (Isomorphic spaces and dimension) Isomorphism: Thm 6.9: (Isomorphic spaces and dimension) Pf: Two finite-dimensional vector space V and W are isomorphic if and only if they are of the same dimension. Elementary Linear Algebra: Section 6.2, p.384

It can be shown that this L.T. is both 1-1 and onto. Thus V and W are isomorphic. Elementary Linear Algebra: Section 6.2, p.384

Ex 12: (Isomorphic vector spaces) The following vector spaces are isomorphic to each other. Elementary Linear Algebra: Section 6.2, p.385

Keywords in Section 6.2: kernel of a linear transformation T: 線性轉換T的核空間 range of a linear transformation T: 線性轉換T的值域 rank of a linear transformation T: 線性轉換T的秩 nullity of a linear transformation T: 線性轉換T的核次數 one-to-one: 一對一 onto: 映成 isomorphism(one-to-one and onto): 同構 isomorphic space: 同構的空間

6.3 Matrices for Linear Transformations Two representations of the linear transformation T:R3→R3 : Three reasons for matrix representation of a linear transformation: It is simpler to write. It is simpler to read. It is more easily adapted for computer use. Elementary Linear Algebra: Section 6.3, p.387

Thm 6.10: (Standard matrix for a linear transformation) Elementary Linear Algebra: Section 6.3, p.388

Pf: Elementary Linear Algebra: Section 6.3, p.388

Elementary Linear Algebra: Section 6.3, p.389

Ex 1: (Finding the standard matrix of a linear transformation) Sol: Vector Notation Matrix Notation Elementary Linear Algebra: Section 6.3, p.389

Check: Note: Elementary Linear Algebra: Section 6.3, p.389

Ex 2: (Finding the standard matrix of a linear transformation) Sol: Notes: (1) The standard matrix for the zero transformation from Rn into Rm is the mn zero matrix. (2) The standard matrix for the zero transformation from Rn into Rn is the nn identity matrix In Elementary Linear Algebra: Section 6.3, p.390

Composition of T1:Rn→Rm with T2:Rm→Rp : Thm 6.11: (Composition of linear transformations) Elementary Linear Algebra: Section 6.3, p.391

Pf: Note: Elementary Linear Algebra: Section 6.3, p.391

Ex 3: (The standard matrix of a composition) Sol: Elementary Linear Algebra: Section 6.3, p.392

Elementary Linear Algebra: Section 6.3, p.392

Inverse linear transformation: Note: If the transformation T is invertible, then the inverse is unique and denoted by T–1 . Elementary Linear Algebra: Section 6.3, p.392

Thm 6.12: (Existence of an inverse transformation) T is invertible. T is an isomorphism. A is invertible. Note: If T is invertible with standard matrix A, then the standard matrix for T–1 is A–1 . Elementary Linear Algebra: Section 6.3, p.393

Ex 4: (Finding the inverse of a linear transformation) Show that T is invertible, and find its inverse. Sol: Elementary Linear Algebra: Section 6.3, p.393

Elementary Linear Algebra: Section 6.3, p.394

the matrix of T relative to the bases B and B': Thus, the matrix of T relative to the bases B and B' is Elementary Linear Algebra: Section 6.3, p.394

Transformation matrix for nonstandard bases: Elementary Linear Algebra: Section 6.3, p.395

Elementary Linear Algebra: Section 6.3, p.395

Ex 5: (Finding a matrix relative to nonstandard bases) Sol: Elementary Linear Algebra: Section 6.3, p.395

Ex 6: Sol: Check: Elementary Linear Algebra: Section 6.3, p.395

Notes: Elementary Linear Algebra, Section 6.3, p.396

Keywords in Section 6.3: standard matrix for T: T 的標準矩陣 composition of linear transformations: 線性轉換的合成 inverse linear transformation: 反線性轉換 matrix of T relative to the bases B and B' : T對應於基底B到B'的矩陣 matrix of T relative to the basis B: T對應於基底B的矩陣

6.4 Transition Matrices and Similarity Elementary Linear Algebra: Section 6.4, p.399

Two ways to get from to : Elementary Linear Algebra: Section 6.4, pp.399-400

Ex 1: (Finding a matrix for a linear transformation) Sol: Elementary Linear Algebra: Section 6.4, p.400

Elementary Linear Algebra: Section 6.4, p.400

Ex 2: (Finding a matrix for a linear transformation) Sol: Elementary Linear Algebra: Section 6.4, p.401

Ex 3: (Finding a matrix for a linear transformation) Sol: Elementary Linear Algebra: Section 6.4, p.401

Thm 6.13: (Properties of similar matrices) Similar matrix: For square matrices A and A‘ of order n, A‘ is said to be similar to A if there exist an invertible matrix P s.t. Thm 6.13: (Properties of similar matrices) Let A, B, and C be square matrices of order n. Then the following properties are true. (1) A is similar to A. (2) If A is similar to B, then B is similar to A. (3) If A is similar to B and B is similar to C, then A is similar to C. Pf: Elementary Linear Algebra: Section 6.4, p.402

Ex 4: (Similar matrices) Elementary Linear Algebra: Section 6.4, p.403

Ex 5: (A comparison of two matrices for a linear transformation) Sol: Elementary Linear Algebra: Section 6.4, p.403

Elementary Linear Algebra: Section 6.4, p.403

Notes: Computational advantages of diagonal matrices: Elementary Linear Algebra: Section 6.4, p.404

Keywords in Section 6.4: matrix of T relative to B: T 相對於B的矩陣 transition matrix from B' to B : 從B'到B的轉移矩陣 transition matrix from B to B' : 從B到B'的轉移矩陣 similar matrix: 相似矩陣